Average complexity of the Best Response Algorithm in Potential Games

In a potential game with $N$ players and action space per player of
size $A$, we show that
the worst case complexity (number of moves) is $N A^{N-1}$, while the average complexity does not depend on $A$ and
equals $log(N) + e^gamma + O(1/N)$.
We also show that the average number of comparisons performed by the
algorithm is $ e^gamma (A-1)(N-1) + O(A)$,
We believe that this result qualifies the Best Response Algorithm as a very efficient
distributed algorithm to compute Nash Equilibria.